A graph is 3-e.c. if, for each triple S of vertices and for each
TÍS, there exists a vertex not in S which is adjacent to
all vertices of T and to no vertices of S \T. The
structure and symmetry of an affine plane provide us with tools for
constructing families of 3-e.c. graphs on the point set of the plane
and for determining when the resulting graphs are non-isomorphic.

A geometric configuration (pq, nk) is a collection of p points
and q straight lines, usually in the Euclidean plane, so that each
point lies on q lines and each line passes through k points. This
talk will discuss configurations with high degrees of symmetry; in
other words, configurations in which the number of symmetry classes of
points and of lines formed by isometries of the plane mapping the
configuration to itself is small. Of particular interest will be
astral configurations, where a (pq, nk) configuration is
astral if it has precisely ë[(k+1)/2] û symmetry
classes of points and ë[(q+1)/2] û symmetry classes
of lines-the fewest possible.

The following problem was posed by Littlewood in 1968. What is the
maximum number of congruent infinite circular cylinders that can be
arranged in R3 so that any two of them are touching? Is it 7? It
was proved by the author in 2005 that this maximum number is at most
24. The talk will also survey models of mutually touching cylinders
and explain the connection of this problem to Gardner's (1959)
mathematical puzzle concerning cigarettes.

P. Soltan (1972) introduced the concept of X-ray numbers of convex
bodies. According to a conjecture of K. Bezdek and T. Zamfirescu
(1991), the X-ray number of any 3-dimensional convex body is at most
6. I give a proof of this conjecture for any convex body with affine
symmetry.

A set in a metric space is called a Cebysev set if it contains
a unique "nearest neighbour" to each point of the space. In this
paper we generalize this notion, defining a set to be Cebysev
relative to another set if every point in the second set has a unique
"nearest neighbour" in the first. We are interested in
Cebysev sets in some hyperspaces over Rn, endowed with the
Hausdorff metric, mainly the hyperspaces of compact sets, compact
convex sets, and strictly convex compact sets.

We present some new classes of Cebysev and relatively
Cebysev sets in various hyperspaces. In particular, we show
that certain nested families of sets are Cebysev; as these
families are characterized purely in terms of containment, without
reference to the semi-linear structure of the underlying metric space,
their properties differ markedly from those of known Cebysev
sets. (A conjectured link with symmetry did not materialize; thus
this paper has become, in this session, something of a lucus a
non lucendo.)

Inspired by Barany's colourful Carathéodory theorem, we introduce a
colourful generalization of Liu's simplicial depth of a point p in
Rd relative to a fixed set S of sample points, i.e., the
number of simplices generate by points in S that contain p. We
prove a parity property and conjecture that the minimum colourful
simplicial depth of any core point in any d-dimensional
configuration is d2+1 and that the maximum is dd+1+1. We
exhibit configurations attaining each of these depths, and apply our
results to the problem of bounding monochrome (non-colourful)
simplicial depth. Independently found recent quadratic lower bounds
by Barany and Matousek and by Stephen and Thomas are also presented.

A hexagon inscribed in a conic determines 60 Pascal lines, and these
generate a remarkable configuration of 146 points and 110 lines.
Although the complete Pascal figure has been studied since the middle
of the 19th century, it is only with today's computer graphics that we
are easily able to see it. For the computer to do its job a
systematic notation is needed; this was provided by J. J. Sylvester
(1844), with improvements from Stanley Payne (1973). Norma Fuller and
I will soon have a web page ready to display the figure. In my talk I
will demonstrate how the viewer will be able to explore the numerous
subconfigurations and see how they fit together to form the whole
figure.

The question which of the seventeen wallpaper groups are represented
in the fabled ornamentation of the Alhambra has been raised and
discussed quite often, with widely diverging answers. Some of the
arguments from these discussions will be presented in detail. This
leads to the more general problem about the validity and meaning of
the answers to such questions. The second part of the presentation
will deals with the approaches to symmetry in ornaments of various
cultures that should replace the mechanical counting of the wallpaper
groups that occur. A more reasonable investigation would deal with
symmetries as they may be considered and understood by the people of
the societies in question. Such an approach to the remarkable orderly
decorations of ancient Peruvian fabrics will be presented.

Can one find n points in dimension m with pairwise distances being
integral (rational)? What is the minimum largest distance (diameter)
for n points, for n points in semi-general position (no m+1
points in a hyperplane), and for n points in general position (no
m+1 points in a hyperplane, no m+2 points on a hypersphere)? What
about a rational box? What about graph drawings in the plane with
straight line edges of integral length?

... a little. In a natural way, the faces of ranks 1 and 2 in a
4-polytope P provide the vertices of a bipartite graph
G. Recently, Asia Weiss and I have examined this
construction when P is a finite, abstract regular (or
chiral) polytope of Schläfli type {3,q,3}. If in this case
P is also self-dual, then G must be
3-transitive (or 2-transitive). Here I discuss further work with
the additional help of Egon Schulte and Tomaz Pisanski. We show
that when P is not self-dual, then G
is no more symmetric then it has right to be. Indeed, G
is a trivalent semisymmetric graph, so that Aut
(G) is transitive on edges but not on vertices.

We will discuss some problems of the following type: Let R a set of
red points and B be a set of black points in the plane all in
general position. Find a simple polygon P with vertex set R such
that the interior of P contains as many points of B as possible.

One of the key problems in the theory of configurations is the
question whether a given combinatorial configuration has a geometric
realization. In the last decade Branko Grünbaum systematically
investigated a more tractable variant of this problem, namely for
which types (vr,bk) do there exist geometric configurations. In a
series of papers he developed a large number of interesting
constructions that produce large configurations from small ones,
thereby giving positive answer in majority of cases. In this talk we
present some of his methods as a theory under the name of
Grünbaum incidence calculus.

This talk presents the complete enumeration of chiral polyhedra in
Euclidean 3-space. Chiral, or irreflexibly regular, polyhedra are
nearly regular polyhedra; their geometric symmetry groups have two
orbits on the flags (regular polyhedra have just one orbit), such that
adjacent flags are in distinct orbits. There are several (very)
infinite families of chiral polyhedra, each either with finite skew
polygonal faces and vertex-figures or with infinite helical faces and
planar vertex-figures. Their geometry and combinatorics are rather
complicated.

Let K be the cubic extension of the finite field F=GF(q), where
Q is any prime power. Being a vector space, K is also an affine
3-space. The mapping from K to F known as the Norm imposes a
metric on K. The unit surface C in K is the set of all elements
of K with norm 1. Using the symmetries of the unit surface, we
cover C with a map of type 3,6 (triangular faces, six at each
vertex). It then follows that C is a torus. The covering map has
considerable symmetry, but is not regular.